qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
1,510 | <p>It's hard to prove a number is transcendental (non-algebraic) yet there are some wonderful examples amongst them like π,e and Liouville's number. What's so special about them? </p>
<p>Are most numbers transcendental?</p>
| Michael Lugo | 143 | <p>Most numbers are transcendental. In particular, the set of algebraic numbers is countable -- basically because there are countably many polynomials, each with countably many roots. (This is a generalization of the argument that the rationals are countable.) But there are uncountably many real numbers.</p>
|
1,450,176 | <p>I would like to evaluate this limit :$$\displaystyle \lim_{x \to \infty} ({x\sin \frac{1}{x} })^{1-x}$$.</p>
<p>I used taylor expansion at $y=0$ , where $x$ go to $\infty$ i accrossed this </p>
<p>problem : ${1}^{-\infty }$ then i can't judge if this limit equal's $1$ , </p>
<p>because it is indeterminate case ,T... | Ron Gordon | 53,268 | <p>I think you have to expand out to the next order term in the Taylor series about infinity. Thus</p>
<p>$$\left ( x \sin{\frac1{x}} \right )^{1-x} = \frac{\displaystyle 1-\frac1{6 x^2}}{\displaystyle\left (1-\frac1{6 x^2} \right )^x}$$</p>
<p>Now</p>
<p>$$\left (1-\frac1{6 x^2} \right )^x = \left[ \left (1-\frac... |
4,620,310 | <p>Prove <span class="math-container">$\lim_{x\to\infty}\sin x/x=0$</span> using definition of a limit of a function.</p>
<p>I know that the definition of a limit of a fuction <span class="math-container">$f(x)$</span> (when <span class="math-container">$x\to\infty$</span> and <span class="math-container">$f(x)$</span... | Lorago | 883,088 | <p>You're a bit wrong with what you need to prove. In particular, since you want the limit to be <span class="math-container">$0$</span>, you want <span class="math-container">$\left\lvert\frac{\sin x}{x}\right\rvert<\varepsilon$</span> and <strong>not</strong> <span class="math-container">$\left\lvert\frac{\sin x}{... |
3,867,197 | <p>Let <span class="math-container">$A$</span> be the following matrix</p>
<p><span class="math-container">$$\left(
\begin{array}{ccc}
1 & 0 & x \\
0 & 1 & y \\
x & y & 1
\end{array}
\right)$$</span></p>
<p>I have to prove that if, at least <span class="math-container">$x+y>\frac{3}{2}$</spa... | player3236 | 435,724 | <p>For <span class="math-container">$x+y>\frac32$</span>:</p>
<p><span class="math-container">$$x^2+y^2=\frac12(x^2+y^2+x^2+y^2)\ge\frac12(x^2+y^2+2xy) =\frac12(x+y)^2>\frac98>1$$</span></p>
|
1,879,673 | <p>I have woven the below incomplete proof of the following claim:</p>
<blockquote>
<p><em>Claim</em>. If $X$ is completely regular and $Y$ is a compactification of $X$,
then there is a unique, continuous, surjective, closed map
$g:\beta\left(X\right)\to Y$ which is the identity on
$X$.</p>
</blockquote>
<p><... | Brian M. Scott | 12,042 | <p>Suppose that $f[X]\subsetneqq Y$, let $U=Y\setminus f[\beta X]$, and fix $y\in U$. Then $U$ is an open nbhd of $y$ in $Y$, and $f[X]$ is dense in $Y$, so $U\cap f[X]\ne\varnothing$. Do you see the contradiction?</p>
<p>You’re right to worry about assuming that $Y\subseteq\beta X$: it need not be true, because — as ... |
4,214,329 | <p>For convenience, let <span class="math-container">$(f(x), g(x))$</span> be a solution to the problem. Now,
<span class="math-container">\begin{align*}
f(x) + g(x) &= f(x)g(x) \\
f(x)g(x) - f(x) - g(x) &= 0 \\
f(x)g(x) - f(x) - g(x) + 1 &= 1 \\
(f(x) - 1)(g(x) - 1) &= 1
\end{align*}</s... | YiFan | 496,634 | <p>You showed it yourself! The only way the product of two polynomials can be constant is if they are themselves constant: otherwise, the degree of the product polynomial would not be zero. Thus <span class="math-container">$(f(x)-1)(g(x)-1)=1$</span> immediately implies <span class="math-container">$f,g$</span> are co... |
145,950 | <p>how can I show that any finite CW-space can embedded into an euclidean space of some dimension? Any help or reference would be greatly appreciated.</p>
| Wlodek Kuperberg | 36,904 | <p>If your finite CW-complex is of topological dimension $n$, then it is an $n$-dimensional compact metric space, thus, by the The Menger-Nöbeling theorem (1932), it can be embedded in ${\mathbb R}^{2n+1}$. In this theorem $2n+1$ is the lowest possible dimension, since there exist $n$-dimensional simplicial complexes t... |
15,235 | <p>More precisely, is there a map of schemes $X$ --> $Y$ such that $f$ gives a homeomorphism between $X$ and a closed subset of $Y$, but the corresponding map on sheaves is not surjective?</p>
| Emerton | 2,874 | <p>Yes, for example if $K \subset L$ is an inclusion fields, then the induced map
Spec $L \to $ Spec $K$ is a homeomorphism (both source and target are single points),
but the induced map on sheaves is the given inclusion of $K$ into $L$, which is
surjective only if $K = L$.</p>
<p>For another example, let $X'\to Y$ ... |
699,383 | <p>I am a non-mathematician who knows some elemententary calculus ans I want to prove that the sequence $(x_n)$ given by</p>
<p>$$
x_n=-\sqrt{n} + n\ln\Big(1+\frac{1}{\sqrt{n}}\Big)
$$</p>
<p>is decreasing. Is there an elegant way to show this?</p>
| Albert | 82,854 | <p>The simplest way might be to expand (n+1)^(1/2) and (n+1)^(-1/2) in the expression for x_(n+1) using the binomial theorem then show that x_(n+1) - x_n is positive.</p>
|
3,753,819 | <p><span class="math-container">$\textbf{Question:}$</span> Let <span class="math-container">$(X,\mathcal{F},\mu)$</span> be an arbitrary measure space. Let <span class="math-container">$\varphi: \mathbb{R} \rightarrow \mathbb{R}$</span> be continuous and satisfy for some <span class="math-container">$K>0$</span>:</... | Kavi Rama Murthy | 142,385 | <p>The result is not true as stated. In the inequality for <span class="math-container">$\phi$</span> we should add 'for <span class="math-container">$|t|$</span> sufficiently large' to make it correct.</p>
<p>First a counterexample: suppose the measure space is finite and <span class="math-container">$\phi (t)=\sqrt {... |
3,079,023 | <p>A vector space with norm <span class="math-container">$\parallel\cdot\parallel$</span> Satisfy for two vectors the following</p>
<p><span class="math-container">$\parallel x+y\parallel=\parallel x\parallel +\parallel y\parallel$</span></p>
<p>i need to proof the fallowing statement</p>
<p><span class="math-contai... | Mustafa Said | 90,927 | <p>To show that the norm is induced by an inner product you can show that the norm satisfies the parallelogram law and then use the polarization identity to recover the norm.</p>
|
3,248,569 | <p>I have the following two parametric equations of lines:</p>
<p><span class="math-container">$$\begin{cases} x = -t + 1 \\ y = t + 3 \\ z = -6t \end{cases} \quad \land \quad \begin{cases} x = 2s + 4 \\ y = -s \\ z = 2s + 1 \end{cases}$$</span></p>
<p>I want to examinate their mutual position, that is, I want to fin... | Hw Chu | 507,264 | <p>The idea is that, if you can show that the ratios of the rectangles are getting closer to <span class="math-container">$\sqrt 3$</span>, then none of them should be similar to each other. So you need to have some way to track <span class="math-container">$\frac{y_n}{x_n}$</span> in each iteration.</p>
<p>If you try... |
85,351 | <p>It has been proven that:</p>
<p>1) if $s$ is a non trivial zero $\rho$ of $\zeta(s)$ then so is $1−s$.</p>
<p>2) $\zeta(s) = 2^s \pi^{s-1} \sin(\frac{\pi s}{2}) \Gamma(1-s) \zeta(1-s)$</p>
<p>3) $ 0 < \Re(\rho) <1$</p>
<p>From this it follows that when $s \to \rho$:</p>
<p>$\displaystyle \lim_{s \to \rho}... | Community | -1 | <p>I do not think there is a nice solution. But the problem can be simplified a bit. First the condition that $f$ is bounded is irrelevant since you can always compose with $\arctan$. Second, the condition that $f$ is a function into $\mathbb{R}$ is not important, and in fact the only thing you need from $f$ is its ker... |
2,526,695 | <p>I've got following sequence formula:
$ a_{n}=2a_{n-1}-a_{n-2}+2^{n}+4$</p>
<p>where $ a_{0}=a_{1}=0$</p>
<p>I know what to do when I deal with sequence in form like this:</p>
<p>$ a_{n}=2a_{n-1}-a_{n-2}$
- when there's no other terms but previous terms of the sequence.
Can You tell me how to deal with this typ... | robjohn | 13,854 | <p>Let $Sa_n=a_{n+1}$ be the shift operator on sequences. Then your equation becomes
$$
\left(1-S^{-1}\right)^2a_n=2^n+4\tag1
$$
where $1-S^{-1}$ is the <a href="https://en.wikipedia.org/wiki/Finite_difference#Forward.2C_backward.2C_and_central_differences" rel="nofollow noreferrer">backward difference operator</a>.</p... |
13,882 | <p>Background: When Ueno builds the fully faithful functor from Var/k to Sch/k he mentions that the variety $V$ can be identified with the rational points of $t(V)$ over $k$. I know how to prove this on affine everything and will work out the general case at some future time.</p>
<p>The question that this got me think... | Pete L. Clark | 1,149 | <p>The following result deals with the case of finite type affine schemes over an arbitrary field <span class="math-container">$k$</span>.</p>
<p>Theorem: Let <span class="math-container">$A$</span> be a finitely generated algebra over a field <span class="math-container">$k$</span>. Let <span class="math-container">$... |
2,967,615 | <p>Given for example 2 functions,<span class="math-container">$\ n^{100} $</span> and<span class="math-container">$\ 2^n$</span>. I know that<span class="math-container">$\ 2^n$</span> grows faster and that therefore there is some<span class="math-container">$\ n$</span> where it will eventually overtake <span class="m... | DeepSea | 101,504 | <p>Solve <span class="math-container">$2^n > n^{100} \iff n\ln 2>100\ln n\iff \dfrac{n}{\ln n} > \dfrac{100}{\ln 2}$</span>. Let <span class="math-container">$n = 2^k$</span>, then <span class="math-container">$ \dfrac{n}{\ln n} = \dfrac{2^k}{k}$</span>,and you solve <span class="math-container">$2^k > 100k... |
2,967,615 | <p>Given for example 2 functions,<span class="math-container">$\ n^{100} $</span> and<span class="math-container">$\ 2^n$</span>. I know that<span class="math-container">$\ 2^n$</span> grows faster and that therefore there is some<span class="math-container">$\ n$</span> where it will eventually overtake <span class="m... | Mark Bennet | 2,906 | <p>One elementary way of proving that exponentials grow faster is to use the binomial expansion.</p>
<p>To illustrate, here <span class="math-container">$(1+1)^r=1+r+\binom r2+\binom r3+\dots$</span>, where we can choose <span class="math-container">$r$</span> large enough for all the terms we need to exist on the rig... |
1,918,071 | <p>Sometimes you will see theorems of the form "Let $H_1, \dots, H_n$. If $A$, then $B$". Sometimes "suppose" or "if" is used instead of "let". Here's an example:</p>
<ol>
<li><p>Let $x\in\mathbb{R}$. If $x\geq 0$, then $|x|=x$.</p></li>
<li><p>Suppose $x\in\mathbb{R}$. If $x\geq 0$, then $|x|=x$.</p></li>
<li><p>If $... | Community | -1 | <p>$e^x (e^x+1) = 2$ does <strong>not</strong> imply that $e^x = 2$ or $e^x+1 = 2$. This type of reasoning only works if the right-hand side is zero. (This is why it's called the <a href="https://en.wikipedia.org/wiki/Zero-product_property" rel="nofollow">zero-product property</a>.)</p>
<p>In general, if $ab = 2$ th... |
2,607,090 | <p>I have a function for which I know:</p>
<p>$f(2) = 2x -3y \\
f(3) = 5x - 6y \\
f(4) = 9x - 10 y \\
f(5) = 14x - 15y$</p>
<p>Assuming that $f$ is a polynomial, how do I find the general expression for $f$? After many minutes of fiddling I eventually found that this general expression works:</p>
<p>$f(N) = \frac{N(... | fleablood | 280,126 | <p>If you are assuming there is a pattern and that </p>
<p>$f(n) = A_n x - B_n y$ then all you are asking is:</p>
<p>If $A_2= 2; A_3 = 5; A_4 = 9; A_5 = 14$ what is $A_n$? And if $B_2= 3;B_3= 6; B_4 = 10; B_5 = 15$ what is $B_n$?</p>
<p>Well, there is no answer as just because something follows a pattern for $4$ num... |
3,290,199 | <p>If I throw a fair dice <span class="math-container">$12$</span> times, the expected number of <span class="math-container">$6$</span> is <span class="math-container">$2$</span> i.e <span class="math-container">$6$</span> is expected to appear <span class="math-container">$2$</span> times when the dice is thrown... | José Carlos Santos | 446,262 | <p>The existence of an antiderivative and being integrable are distinct (although related) concepts.</p>
<p>Take<span class="math-container">$$\begin{array}{rccc}f\colon&[0,1]&\longrightarrow&\mathbb R\\&&x\mapsto&\begin{cases}x^2\sin\left(\frac1{x^2}\right)&\text{ if }x>0\\0&\text{ ... |
2,244,423 | <p>The function given is $f(x) = \sqrt[3]{{x}^2(2-x)}$.</p>
<p>Can anybody help me to find all asymptotes of this function. I know it doesn't have a vertical asymptote and I know that it's horizontal asymptote is $\sqrt[3]{-1}$, but I don't know how to find asymptote of the slope.</p>
<p>I'd prefer if someone could h... | Minz | 435,601 | <p>$\sqrt[3]{x^2(2-x)}=-x\sqrt[3]{1-\frac2x}=\boxed{\text{via Taylor}}=-x(1-\frac2{3x}+o(\frac1x))=-x+\frac23+o(1)$</p>
<p>As $\sqrt[3]{x^2(2-x)}- (-x+\frac23)$ tends to $0$ the asymptotes at $-\infty$ and $+\infty$ are $y=-x+\frac23$ according to definition.</p>
<p>Alternative approach is standart:
firstly search ... |
4,037,295 | <p>The Cauchy Schwarz inequality says
<span class="math-container">$$
(ax+by+cz)^2 = (a^2+b^2+c^2)(x^2+y^2+z^2).
$$</span></p>
<p>I found that there is a kind of analogous inequality for <span class="math-container">$(ax+by+cz)^n$</span>
<span class="math-container">$$
(x+y+z)^n \leq 3^n(x^n+y^n +z^n).
$$</span>
if I r... | WimC | 25,313 | <p>The factor can be improved to <span class="math-container">$3^{n-1}$</span>. For odd <span class="math-container">$n$</span> you have to assume probably that <span class="math-container">$x,y,z \geq 0$</span>. Then this is <a href="https://en.wikipedia.org/wiki/Jensen%27s_inequality" rel="nofollow noreferrer">Jensen... |
122,293 | <p>Let's consider all possible permutations of N numbers. Suppose for each permutation we calculate the sum of absolute differences between consecutive elements. Thus, for (1,2,3) one would have abs(1-2)+abs(2-3)=2. Is it possible to obtain a distribution of such sums for given N? For instance, for N=3 one would have 3... | Douglas Zare | 2,954 | <p>Andrew King pointed out that $E[X]$ is $(n^2-1)/3$. </p>
<p>So, to calculate the variance $E[X^2] - E[X]^2$ we need to find $E[X^2]$.</p>
<p>Let $\delta(i) = |\pi(i+1)-\pi(i)|$, so $X = \sum_{i=1}^{N-1} \delta(i)$.</p>
<p>$$\begin{eqnarray}X^2 &=& \bigg(\sum_{i=1}^{N-1} \delta(i)\bigg)^2 \newline &=&a... |
1,683,238 | <p>Let $[x]$ denote the fractional part of x. I'm quite lost about how to solve this problem. I suspect the solution is elementary, but all I can determine is that $x\notin\Bbb{Q}$.</p>
| John Wayland Bales | 246,513 | <p>Have you tried proof by contradiction? Suppose there is a positive $\alpha$ less than 1 such that for all positive $x$ less than 1 there exists an $n\in\mathbb{N}$ such that $\alpha^n\ge[nx]$. Perhaps for $x$ some function of $\alpha$ this leads to a contradiction? Such as perhaps $x=1-\alpha$ although that is just ... |
237,838 | <p>The data are for the model $T(t) = T_{s} - (T_{s}-T_{0})e^{-\alpha t}$,
where $T_0$ is the temperature measured at time 0, and $T_{s}$ is the temperature at time $t=\infty$, or the environment temperature. $T_{s}$ and $\alpha$ are parameters to be determined.</p>
<p>How can I fit my data against this model? I'm try... | Community | -1 | <p>Gradient descent might be overkill. </p>
<p>For convenience, use a temperature scale translated so that $T_0=0$ and the model is</p>
<p>$$T(t)=T_s(1-e^{-\alpha t}).$$</p>
<p>You want to minimize </p>
<p>$$E=\sum_i(T_i-T_s(1-e^{-\alpha t_i}))^2.$$</p>
<p>Setting an arbitrary value for $\alpha$, the least-squares... |
356,574 | <blockquote>
<p>$$\aleph_2^{\aleph_0}=\aleph_2$$</p>
</blockquote>
<p>Appreciate your help</p>
| Asaf Karagila | 622 | <p>One can use Hausdorff's formula (if it is for their disposal),</p>
<blockquote>
<p>$$\aleph_{\alpha+1}^{\aleph_\beta}=\aleph_\alpha^{\aleph_\beta}\cdot\aleph_{\alpha+1}$$</p>
</blockquote>
<p>From there we have: $$\aleph_2^{\aleph_0}=\aleph_1^{\aleph_0}\cdot\aleph_2=\left(2^{\aleph_0}\right)^{\aleph_0}\cdot\alep... |
3,370,076 | <p>The total mechanical energy is conserved when a ball is dropped from a height of 4.00 <span class="math-container">$\mathit{m}$</span>, and it makes a elastic collision with the ground. Assuming no non-conservative forces are acting find the period of the ball. g of course is 9.81.</p>
<p><span class="math-containe... | Danny Pak-Keung Chan | 374,270 | <p>It is trivial. Let <span class="math-container">$x_{1},x_{2}\in G$</span>. If <span class="math-container">$\alpha(x_{1})=\alpha(x_{2})$</span>,
then <span class="math-container">$ax_{1}=ax_{2}\Rightarrow a^{-1}(ax_{1})=a^{-1}(ax_{2})\Rightarrow x_{1}=x_{2}$</span>.
Therefore <span class="math-container">$\alpha$</s... |
301,264 | <p>Note: There is another question of the same title, but it is different and asks for group theory prerequisites in algebraic topology, while i want the topology prerequisites. </p>
<p>I am a physics undergrad, and I wish to take up a course on Introduction to Algebraic Topology for the next sem, which basically teac... | sponsoredwalk | 43,502 | <p>Chapter 1 of Hatcher corresponds to chapter 9 of Munkres. <a href="http://www.ictp.tv/diploma/search07-08.php?activityid=MTH&course=Topology">These</a> topology video lectures (syllabus <a href="http://diploma.ictp.it/courses/math/topology-top">here</a>) do chapters 2, 3 & 4 (topological space in terms of op... |
3,406,106 | <p>Want to prove rigorously (if possible, since I was not able to think of any counter-example) that <span class="math-container">$\lim_{x\to a} f(x)$</span> exists <span class="math-container">$\implies \lim_{x\to a} f(x^2)$</span> exists. (I also have a feeling that the limits equate.)</p>
<p>I started with the <s... | user | 505,767 | <p>It is always true for <span class="math-container">$a=0$</span> and <span class="math-container">$a=1$</span> otherwise just assume a discontinuous function at <span class="math-container">$x=a^2$</span>.</p>
|
3,522,736 | <p>I've been messing around with trying to negate this statement using DeMorgans laws and I keep ending up with incorrect answers such as (~p or ~q) and ~r. If someone could help me with the negation of compound statements. </p>
<p>Thank you.</p>
| Martin Argerami | 22,857 | <p>Since <span class="math-container">$x>0$</span>, you have <span class="math-container">$1+x>0$</span>, and <span class="math-container">$\frac1{1+x}>0$</span>. </p>
<p>And <span class="math-container">$1+x>1$</span>, so <span class="math-container">$\frac1{1+x}<1$</span>. </p>
<p>Equality is not ach... |
3,669,539 | <p>I’ve been looking at non-trivial solutions of ODEs. I found one and have problems with it.</p>
<p><span class="math-container">$y’(x)=\frac{1}{4}\sqrt{y(x)}$</span></p>
<p><span class="math-container">$y(0)=0$</span></p>
<p>I know the one of the solutions of this ODE is</p>
<p><span class="math-container">$y(x) ... | Bernard | 202,857 | <p>This is an antiderivative problem. The fundamental theorem of integral calculus asserts that
<span class="math-container">$$y(x)=\int_0^x\tfrac14\sqrt{ t\mkern1mu}\mkern3mu\mathrm dt=\tfrac14\tfrac23t^{3/2}\bigg|_0^x{}=\tfrac16 x^{3/2}.$$</span></p>
|
497,422 | <p>Which is bigger: $a$ or $a^2$ and what is the proof of that?</p>
<p>I'm kinda stuck and because there are cases where $a$ is bigger and other cases where $a^2$ is bigger.</p>
| mrf | 19,440 | <p>Hint: Factorize $a^2 - a$. When is this number greater than or smaller than $0$?</p>
|
1,106,325 | <p>In <a href="https://www.encyclopediaofmath.org/index.php/Codimension" rel="nofollow noreferrer">this article about codimension</a> there is the following remark: </p>
<p>The codimension of a subspace $L$ of a vector space $V$ is equal to the dimension of any complement of $L$ in $V$, since all complements have the... | user1801328 | 147,676 | <p>The most general statement of orthogonality I've seen is in a Banach space. If you have a Banach space $X$ and denote it's dual by $X^{*} $ then, for $V \subseteq X$, the complement of $V $ is defined as all $x^{*} \in X^{*}$ s.t for every $v \in V $, $x^{*}(v)=0$. </p>
|
881,520 | <p>Take half a square with side length $1$.
The resulting right-angled triangle ABC
has two angles of $45^\circ$.
By Pythagoras’ theorem, the hypotenuse
AC has length $\sqrt{2}$. Applying the definitions on the previous page gives the values in the table below. that $\sin 30^\circ= \frac{1}{2}$</p>
<p>Sorry I cannot ... | Henry | 6,460 | <p>Take an equilateral triangle of side $1$. </p>
<p>Bisect it through a vertex and the midpoint of the opposite side. </p>
<p>You now have two right-angled triangles with angles $30^\circ, 60^\circ, 90^\circ$ with the edge opposite the $30^\circ$ of length $\frac12$ and hypotenuse $1$. So $$\sin (30^\circ)=\frac... |
550,659 | <blockquote>
<p>A space <span class="math-container">$X$</span> is locally metrizable if each point <span class="math-container">$x$</span> of <span class="math-container">$X$</span> has a neighborhood that is metrizable in the subspace topology. Show that a compact Hausdorff space <span class="math-container">$X$</spa... | Berci | 41,488 | <p>Of course, if $X$ is metrizable then it is also locally metrizable. </p>
<p>For the other direction, you could really follow the hint. Supposed that $X$ is locally metrizable, there exists an open cover $\bigcup_{x\in X}U_x$ of $X$ where $U_x$ is a metrizable neighborhood of $x$. Because of compactness, it has a fi... |
550,659 | <blockquote>
<p>A space <span class="math-container">$X$</span> is locally metrizable if each point <span class="math-container">$x$</span> of <span class="math-container">$X$</span> has a neighborhood that is metrizable in the subspace topology. Show that a compact Hausdorff space <span class="math-container">$X$</spa... | user558252 | 558,252 | <p>[Hint: Using normality/regularity]</p>
<p>The space $X$ is compact Hausdorff, so it is normal/regular.</p>
<p>Now for each $x\in X$, there is a metrizable open neighborhood $U_{x}$ of $x$, for this $U_{x}$, by regularity, there is an open neighborhood $V_{x}$ of $x$ such that
$x\in V_x \subset V_{x}^{-} \subset U... |
2,564,217 | <p>For a project I'm doing, I'm wrapping an led strip light around a tube. The tube is 19mm in diameter and 915mm tall. I'm going to coil the led strip around the tube from top to bottom and the strip is 8mm wide, so the coils will be 8mm apart. How long does the led strip need to be to fully cover the tube?</p>
<p>Th... | Graham Kemp | 135,106 | <blockquote>
<p>I drew a 6x6 grid and tried labeling all the possible outcomes followed by the number of times they occur, but I feel like I got no where. I know approach is very wrong.</p>
</blockquote>
<p>The approach is correct. The table should look as follows, just complete it.</p>
<p>$$\begin{array}{:c|c:c:c... |
2,956,791 | <p>There is an equation here:
<span class="math-container">$$\sqrt{x+1}-x^2+1=0$$</span>
Now we want to write the equation <span class="math-container">$f(x)$</span> like <span class="math-container">$h(x)=g(x)$</span> in a way that we know how to draw h and g functions diagram.
Then we draw the h and g function diagra... | Dr. Sonnhard Graubner | 175,066 | <p>Hint: Write your equation in the form
<span class="math-container">$$\sqrt{x+1}=(x+1)(x-1)$$</span></p>
|
2,142,042 | <p>how would you use induction to prove this:</p>
<p>$\sin(x)-sin(3x)+sin(5x)-...+(-1)^{(n+1)}sin[(2n-1)x] = \frac{(-1)^{(n+1)}sin2nx}{2cosx} $</p>
<p>I know how you assume its true for n=k, and then prove for n=k+1, but I get to </p>
<p>Left Hand Side: $\frac{(-1)^{(k+1)}sin2kx}{2cosx}+(-1)^{k+2}sin[(2k+1)x]$ but I... | Community | -1 | <p>First pull the constant $a$ and $b$ away by a scaling of the variable, and get rid of the square with $ax^2/b\to t$.</p>
<p>To a constant factor, you now have</p>
<p>$$\int_0^\infty\frac{t^{p-1}}{(t+1)^{p+q}}dt.$$</p>
<p>Now you convert to the classical Beta form by $u:=t/(t+1)$,</p>
<p>$$\int_0^1u^{p-1}(1-u)^{q... |
1,050,917 | <p>I have a problem that I have to solve. I need to find center of the circle containing the point $(x,y)$. The point is $x=2,y=3$ with radius $r=3$. I need to find the center of circle. Is there equation for that? I use this equation.<br>
$$(x-h)^2+(y-k)^2=r^2$$
How I can find $h$ and $k$ for the center of circle if ... | Archis Welankar | 275,884 | <p>Acoording to you $(x,y)$ is the centre with radius $3$ . So by the distance formula $(x-2)^2+(y-3)^2=3^2=9$. Then by plugging in any value for y you get a corresponding value for x. There are 2 unknowns and only 1 equation. So we need another parameter to solve for a specific point. Two, of the infinitely many, answ... |
135,936 | <p>I need this one result to do a problem correctly.</p>
<p>I want to show that for any $b \in \mathbb{C}$ and $z$ a complex variable:</p>
<p>$$ |z^2 + b^2| \geq |z|^{2} - |b|^{2}$$ </p>
<p>My attempts have only led me to conclude that </p>
<p>$$ |z^2 + b^2| > \frac{|z|^{2} + |b|^{2}}{2}$$ </p>
| martini | 15,379 | <p>$$ |z|^2 = |z^2| = |z^2 + b^2 - b^2| \le |z^2 + b^2| + |b|^2 $$
and your inequality follows. I suppose you are missing an $|\cdot|$ arround the $b$? For you cannot compare complex numbers well.</p>
|
4,532,279 | <p>Let <span class="math-container">$k$</span> be an algebraically closed field and <span class="math-container">$X$</span> be an affine surface over <span class="math-container">$k$</span>. Suppose <span class="math-container">$\phi: \mathbb{A}^1 \to X$</span> is a non-constant morphism. Then we know that <span class=... | Qiaochu Yuan | 232 | <p>Conrad's suggestion in the comments works great and is probably the most low-tech solution. Here are some others. Write <span class="math-container">$z = e^{i \theta}$</span>, so the question is to show that <span class="math-container">$z^n + z^{-n}$</span> is a polynomial in <span class="math-container">$z + z^{-1... |
4,532,279 | <p>Let <span class="math-container">$k$</span> be an algebraically closed field and <span class="math-container">$X$</span> be an affine surface over <span class="math-container">$k$</span>. Suppose <span class="math-container">$\phi: \mathbb{A}^1 \to X$</span> is a non-constant morphism. Then we know that <span class=... | GEdgar | 442 | <p>Prove more: For all positive integers <span class="math-container">$n$</span>,
<span class="math-container">$$
\cos(n\theta) = T_n(\cos(\theta)),
\qquad \sin(n\theta) = \sin(\theta) U_{n-1}(\cos\theta)
$$</span>
where <span class="math-container">$T_n(x)$</span> and <span class="math-container">$U_{n-1}(x)$</span> a... |
14,385 | <p>I have always taught my students that the <span class="math-container">$y$</span>-intercept of a line is the <span class="math-container">$y$</span>-coordinate of the point of intersection of a line with the <span class="math-container">$y$</span>-axis, that is, for the line given by the equation <span class="math-c... | Xander Henderson | 8,571 | <p>Some time spent with the Google does not seem to turn up very much. The best answer that I can find seems to be from the Common Core Standards, and is ambiguous, at best. On page 69 of the <a href="https://www.engageny.org/resource/new-york-state-p-12-common-core-learning-standards-for-mathematics" rel="noreferrer... |
14,385 | <p>I have always taught my students that the <span class="math-container">$y$</span>-intercept of a line is the <span class="math-container">$y$</span>-coordinate of the point of intersection of a line with the <span class="math-container">$y$</span>-axis, that is, for the line given by the equation <span class="math-c... | Edwin F. Sampang | 14,866 | <p>I think y-intercept is a directed line segment. It is a vector in which it has a magnitude and direction. For example for the line y=2x-8, the y-intercept must be -8 meaning it is 8 units downward from the origin. The point involved is (0,-8) but it is not the y-intercept. It is only the point where the line crosses... |
3,969,598 | <p>Let <span class="math-container">$f: \mathbb{R} \rightarrow \mathbb{R}$</span> a monotonically increasing function and <span class="math-container">$A \subset \mathbb{R}$</span> where <span class="math-container">$A \neq \emptyset$</span> and boundend.</p>
<p>i) If f is continuous function, prove that <span class="... | Ross Millikan | 1,827 | <p><span class="math-container">$$s=r-\sqrt{r^2-y^2}\\
\sqrt{r^2-y^2}=s-r\\
r^2-y^2=s^2-2sr+r^2\\
2sr=s^2+y^2\\
r=\frac{s^2+y^2}{2s}$$</span>
As we squared in the third line we may have introduced an extraneous solution, so the solution needs to be checked back into the original equation</p>
|
3,692,877 | <p>Consider the following expression in three variables, <span class="math-container">$0 \leq p,s \leq 1$</span> and <span class="math-container">$n >0$</span></p>
<p><span class="math-container">$$S_{n, p, s} = \sum_{k=0}^n {n \choose k} p^k (1-p)^{n-k} e^{-s(k - np)^2}$$</span></p>
<p>If <span class="math-contai... | Oliver Díaz | 121,671 | <p>I don't think there is a closed form. As for when <span class="math-container">$n\rightarrow\infty$</span>, it seems that the limit is <span class="math-container">$0$</span> for <span class="math-container">$s>0$</span>. Here is a sketch of the proof, hoping there are no embarrassing typos.</p>
<p><span class="... |
33,215 | <p>There is a huge debate on the internet on the value of <span class="math-container">$48\div2(9+3)$</span>.</p>
<p>I believe the answer <span class="math-container">$2$</span> as I believe it is part of the bracket operation in BEDMAS. <a href="https://www.mathway.com" rel="nofollow noreferrer">Mathway</a> yields the... | Community | -1 | <p>There is no order difference between implicit and explicit multiplications. <a href="http://www.purplemath.com/modules/orderops2.htm" rel="noreferrer">Purplemath</a> suggests that implied multiplication outside of parentheses also gets parenthetical order priority over all other multiplication(division). So they wou... |
33,215 | <p>There is a huge debate on the internet on the value of <span class="math-container">$48\div2(9+3)$</span>.</p>
<p>I believe the answer <span class="math-container">$2$</span> as I believe it is part of the bracket operation in BEDMAS. <a href="https://www.mathway.com" rel="nofollow noreferrer">Mathway</a> yields the... | Justin | 2,533 | <p>It's ambiguous, there is not one right answer in this case, other than possibly that it is undefined. You may have
$$\frac{48}{2(9+3)} = 2$$<br>
or<br>
$$\frac{48}{2}(9+3) = 288$$ </p>
<p>Therefore, there is no point in debating this. </p>
<p>Note that the reason you are get different answers from mathway a... |
677,393 | <p>I have to solve an introductory counting principles problem, It goes like this:</p>
<blockquote>
<blockquote>
<p>Twelve people travel in three cars, with four people in each car. Each car is driven by its owner. Find the number of ways in which the remaining nine people may be allocated to the cars. <strong>(... | ABcDexter | 128,946 | <p>The answer should be $(^9C_3 * ^6C_3 * ^3C_3)$ = 1680. <br>
As first we choose 3 out of 9 and then 3 Out Of remaining 6 and lastly 3 out of 3. <br></p>
<p>The mistake I did was to take extra care of the dissimilarity of the cars which caused the over-addition.</p>
<p><br>
Explanation
The correct answer is : numbe... |
677,393 | <p>I have to solve an introductory counting principles problem, It goes like this:</p>
<blockquote>
<blockquote>
<p>Twelve people travel in three cars, with four people in each car. Each car is driven by its owner. Find the number of ways in which the remaining nine people may be allocated to the cars. <strong>(... | Kiran | 82,744 | <p>Correct answer is ${9\choose3}\times{6\choose3}$.</p>
<p>$3!$ is not needed as it is already factored in ${9\choose3}\times{6\choose3}$. It's easy to understand if we try with small examples.</p>
|
109,569 | <p>Let's say I have a complex valued matrix $\begin{pmatrix}1+I&2+2I&3+3I\\4+4I&5+5I&6+6I\end{pmatrix}$ represented by a list:</p>
<pre><code> list = {{1 + I, 2 + 2 I, 3 + 3 I}, {4 + 4 I, 5 + 5 I, 6 + 6 I}}
</code></pre>
<p>I know how to plot each point of the matrix on the complex plane:</p>
<pre><c... | QuantumDot | 2,048 | <p>I came up with another way to do this:</p>
<pre><code>expr /. b[a_]*c_f :>
Block[{$patternMatched},
With[{result = c /. (d[a] :> ($patternMatched = True; e))},
result /; $patternMatched]]
</code></pre>
<p>Once the outer pattern matches (<code>b[a_]*c_f</code>), a boolean is set up <code>$patternM... |
1,273,441 | <p>In Contests in Higher Mathematics: Miklos Schweitzer Competitions,
1962-1991 by Gabor J Szekely, problem F.57 there is the study of
$f~:~[0,\infty)\to (0,\infty)$ such that: $\exists c>0, \forall x>0$,
$f'(x)=cf(x+1)$.</p>
<p>It states, without proof, that for this equation has a solution if and only if
$c\le... | achille hui | 59,379 | <p>I'm going to prove the "only if" part of the problem. i.e. If $f : [0,\infty) \to (0,\infty)$ is a solution of
the equation</p>
<p>$$f'(x) = c f(x+1),\quad c > 0\tag{*1}$$</p>
<p>then $c \le \frac{1}{e}$, the other direction is leave as an exercise.</p>
<p>For any solution $f(x)$ of the problem, let $\displays... |
309,380 | <p>Let me sum up my - hopefully correct - understanding of the <a href="https://en.wikipedia.org/wiki/Travelling_salesman_problem" rel="nofollow noreferrer">travelling salesman problem</a> and <a href="https://en.wikipedia.org/wiki/Complexity_class" rel="nofollow noreferrer">complexity classes</a>. It's about <a href="... | usul | 29,697 | <p>I think the important distinction that may illuminate your question is between a proof that a particular instance $(W,L)$ belongs to the language TSP, and a proof that a particular algorithm for TSP is correct.</p>
<p>Consider the nondeterministic-Turing-machine $A$ that guesses a path $\pi$ and returns true if the... |
117,500 | <p>How would you go about finding the conjugacy classes of the nonabelian group of order 21, $G:=\left\langle x,y | x^7=e=y^3, y^{-1}xy=x^2\right\rangle$?</p>
| Muhammad Ashraf Rather | 234,522 | <p>Here order of the group is $21= 3 \times 7$ . It is easy to see $3$ divides $7-1=6$. Thus, we have (upto isomorphism)two groups of order $21$. One of them is cyclic $\mathbb{Z}_{21}$ and other is non-abelian. This non-abelian is generated by two elements say $a$ and $b$ such that $|a|=3$ and $|b|=7$ and $ba=ab^r$ wh... |
2,292,015 | <p>The question is:
the first three terms of an arithmetic series $c_{n}$ are
$$a(1+b), a(1+3b),a(1+5b)$$
I needed to find the common difference in terms of $a$ and $b$ and then find the expression for $c_{n}$.</p>
<p>The final part I struggled with where I have to find $a$ and $b$ and the information given is
$$c_{5... | G Tony Jacobs | 92,129 | <p>A function from $\mathbb{R}$ to $\mathbb{R}^3$ generally looks like a path in space. In this case, the path begins at the point $(0,0,0)$ and spirals upward with an increasing radius, basically winding around an up-ward opening cone.</p>
<p>A good way to "see" such a function is to look at projections: Ignore the $... |
316,500 | <p>Let $ A $ be a commutative unital Banach algebra that is generated by a set $ Y \subseteq A $. I want to show that $ \Phi(A) $ is homeomorphic to a closed subset of the Cartesian product $ \displaystyle \prod_{y \in Y} \sigma(y) $. Moreover, if $ Y = \{ a \} $ for some $ a \in A $, I want to show that the map is ont... | Community | -1 | <p>Note that $\Phi (A)$ is compact in the w$^*$-topology. Also, $\prod \sigma(y)$ is compact Hausdorff in the product topology. For the map $f$ you defined, note that $Ker f = \{0\}$ since $Y$ generates $A$.</p>
<p>To prove continuity, take a net $\{\phi_\alpha\}_{\alpha \in I}$ in $\Phi(A)$, such that $\phi_\alpha ... |
316,500 | <p>Let $ A $ be a commutative unital Banach algebra that is generated by a set $ Y \subseteq A $. I want to show that $ \Phi(A) $ is homeomorphic to a closed subset of the Cartesian product $ \displaystyle \prod_{y \in Y} \sigma(y) $. Moreover, if $ Y = \{ a \} $ for some $ a \in A $, I want to show that the map is ont... | Haskell Curry | 39,362 | <p><strong>The mapping $ f $ is well-defined.</strong></p>
<p>Given each $ \phi \in \Phi(A) $, we must have $ \phi(a) \in \sigma(a) $ for all $ a \in A $. Indeed, as $ \phi(a - \phi(a) \cdot \mathbf{1}_{A}) = 0 $, we see that $ a - \phi(a) \cdot \mathbf{1}_{A} $ is not invertible, or equivalently, $ \phi(a) \in \sigma... |
514,338 | <p>Okay so my algebra knowledge is pretty guff..</p>
<p>I am taking a control systems class and pretty much all the questions I am expected to revise, are about doing this algebraic manipulation and I don't know what steps the tutor is taking to do it..</p>
<p>Okay here goes..</p>
<p>If the transfer function of a sy... | Mark Bennet | 2,906 | <p>Note that $$\frac {\frac 3{20s+1}}{\frac3{20s+1}+1}=\frac {\frac 3{20s+1}}{\frac3{20s+1}+1}\cdot\frac {20s+1}{20s+1}=\frac{3}{3+20s+1}=\frac 3{20s+4}$$</p>
|
514,338 | <p>Okay so my algebra knowledge is pretty guff..</p>
<p>I am taking a control systems class and pretty much all the questions I am expected to revise, are about doing this algebraic manipulation and I don't know what steps the tutor is taking to do it..</p>
<p>Okay here goes..</p>
<p>If the transfer function of a sy... | user91011 | 91,011 | <p>\begin{align}
\frac{\frac{3}{20s+1}}{\frac{3}{20s+1}+1} &= \frac{{20s+1}}{{20s+1}}\frac{\frac{3}{20s+1}}{\frac{3}{20s+1}+1}\\
& = \frac{\frac{3(20s+1)}{20s+1}}{\frac{3(20s+1)}{20s+1}+(20s+1)}\\
& = \frac{3}{3+(20s+1)}.
\end{align}</p>
|
1,295,259 | <p>How to prove that;</p>
<blockquote>
<p>$a^{|G|}=e$ if a $\in G $</p>
</blockquote>
<p>if $G$ is a finite group and $e$ is its identity.</p>
<p>I think this could be done through pigeonhole principle but I don't want to use the Lagrange theorem.</p>
<p>How should I start?</p>
| Dietrich Burde | 83,966 | <p>I only know a proof without Lagrange (or Lagrange in disguise) for an abelian group. Suppose that $G=\{ a_1,\ldots ,a_n\}$, and set $g:=a_1a_2\cdots a_n$. Then for every $x\in G$ the map $a_i\mapsto xa_i$ is a permutation of $G$, so that
$$
g=(xa_1)(xa_2)\cdots (xa_n)=x^na_1a_2\cdots a_n=x^ng.
$$
This implies $x^{\m... |
1,041,731 | <p>I want to prove that if $A$ in an infinite set, then the cartesian product of $A$ with 2 (the set whose only elements are 0 and 1) is equipotent to $A$.</p>
<p>I'm allowed to use Zorn's Lemma, but I can't use anything about cardinal numbers or cardinal arithmetic (since we haven't sotten to that topic in the course... | Rodrigo de Azevedo | 339,790 | <p>Visual inspection tells us that matrix <span class="math-container">$\rm A$</span> is a <a href="http://en.wikipedia.org/wiki/Companion_matrix" rel="nofollow noreferrer">companion matrix</a> and that <span class="math-container">$1$</span> is an eigenvalue of <span class="math-container">$\rm A$</span>. Hence, the c... |
83,965 | <p>When students learn multivariable calculus they're typically barraged with a collection of examples of the type "given surface X with boundary curve Y, evaluate the line integral of a vector field Y by evaluating the surface integral of the curl of the vector field over the surface X" or vice versa. The trouble is t... | J.C. Ottem | 3,996 | <p>The <a href="http://www.math.duke.edu/~wka/math204/fixed.pdf">proof of Brower's fixed point theorem</a> using Stokes' theorem is a nice application I think.</p>
|
76,360 | <p>I have a string of characters, like</p>
<pre><code>"CDABOZPVRYXSWQEGNILUTHMKJF"
</code></pre>
<p>and want to convert it to a string in which the character at position <code>p</code> is bold. </p>
<p>After doing this I want to leave the result in a table that will be displayed in <code>TableForm</code> and then us... | Mr.Wizard | 121 | <p>This question is related to at least:</p>
<ul>
<li><a href="https://mathematica.stackexchange.com/q/7732/121">Highlighting text with StringReplacePart but also using Style, Subscript</a></li>
<li><a href="https://mathematica.stackexchange.com/q/10990/121">How to join two Style[]d strings</a></li>
</ul>
<p>Fortunat... |
76,360 | <p>I have a string of characters, like</p>
<pre><code>"CDABOZPVRYXSWQEGNILUTHMKJF"
</code></pre>
<p>and want to convert it to a string in which the character at position <code>p</code> is bold. </p>
<p>After doing this I want to leave the result in a table that will be displayed in <code>TableForm</code> and then us... | kglr | 125 | <pre><code>srF = StringReplacePart[#, ToString[Style[StringTake[#, {#2}], ##3], StandardForm], {#2, #2}] &;
str = "CDABOZPVRYXSWQEGNILUTHMKJF";
srF[str, #, Red, Bold, 16] & /@ {3,9}
</code></pre>
<p><img src="https://i.stack.imgur.com/mwEAu.png" alt="enter image description here"></p>
<p>Note: Copy as LateX ... |
3,999,699 | <p>I want to show that <span class="math-container">$a_{n} = \sqrt{n}$</span> is not a bounded sequence.</p>
<p>Definition: We say that a sequence is bounded if it is bounded above and below. A sequence <span class="math-container">$a_{n}$</span> is bounded above if there exists <span class="math-container">$C$</span> ... | Ted Shifrin | 71,348 | <p>The sloppy reasoning occurs with the right triangle you wrote. You must be careful about the <em>range</em> (values) of the arcsec function. <span class="math-container">$y=\text{arcsec}(x)$</span> can lie <em>either</em> in <span class="math-container">$[0,\pi/2)$</span> <em>or</em> in <span class="math-container">... |
1,083,277 | <p>$a,b,c \in \mathbb{R}$ and $a+b+c=0$.
Prove that: $8^{a}+8^{b}+8^{c}\geqslant 2^{a}+2^{b}+2^{c}$</p>
<p>I think that $2^{a}.2^{b}.2^{c}=1$, but i don't know what to do next</p>
| DeepSea | 101,504 | <p><strong>Hint:</strong> $f(x) = x^3 - x$, $g(x) = \dfrac{x^3}{9} - x$ $\text{ increases}$, $x \geq 3$ and Jensen's inequality !</p>
|
1,083,277 | <p>$a,b,c \in \mathbb{R}$ and $a+b+c=0$.
Prove that: $8^{a}+8^{b}+8^{c}\geqslant 2^{a}+2^{b}+2^{c}$</p>
<p>I think that $2^{a}.2^{b}.2^{c}=1$, but i don't know what to do next</p>
| math110 | 58,742 | <p>Let $x=2^a,y=2^b,z=2^c$. Then $x,y,z>0$ and
$$ xyz=2^{a+b+c}=1.$$
By Holder's inequality
$$(x^3+y^3+z^3)(1+1+1)(1+1+1)\ge (x+y+z)^3.$$
Therefore,
$$x^3+y^3+z^3\ge\dfrac{(x+y+z)^3}{9}\ge (x+y+z)$$
because use AM-GM inequality
$$(x+y+z)^2\ge (3\sqrt[3]{xyz})^2=9.$$</p>
|
3,364,404 | <blockquote>
<p>If <span class="math-container">$X\sim \operatorname{Binom}(n,p)$</span> and <span class="math-container">$Y\sim \operatorname{Ber}\left(\frac Xn\right)$</span>, then find <span class="math-container">$E[X\mid Y]$</span>.</p>
</blockquote>
<p>Is there a name for such a random variable <span class="ma... | drhab | 75,923 | <p>In this answer for convenience <span class="math-container">$q:=1-p$</span> and <span class="math-container">$Z\sim\mathsf{Bin}\left(n-1,p\right)$</span>.</p>
<p>Working out: <span class="math-container">$$P\left(Y=1\right)=\sum_{k=0}^{n}P\left(Y=1\mid X=k\right)P\left(X=k\right)$$</span>
we find <span class="math-... |
650,395 | <p>I am given generating functions $f(x)= \frac{x}{1-x}$ or $f(x)=\frac{1}{1+x^{2}}$ or $f(x)=\frac{1}{x^2-5x+6}$ and I am obliged to write sequence which are generated by this functions. What is the fastest algorithm to solve these problems? I have problem with even starting. I will be glad if anyone would be so nice ... | Adi Dani | 12,848 | <p>$$\frac{x}{1-x}=x\frac{1}{1-x}=x\sum_{n=0}^{\infty}x^n=\sum_{n=0}^{\infty}x^{n+1}=\sum_{n=1}^{\infty}x^{n}$$
$$\frac{1}{1+x^2}=\frac{1}{1-(-x^2)}=\sum_{n=0}^{\infty}(-x^2)^n=\sum_{n=0}^{\infty}(-1)^nx^{2n}$$</p>
|
2,143,510 | <p>NOTE: There are some other similar questions, but I got a negative answer to this question from my proof. Please find out the errors in my reasoning. </p>
<p>$\mathbf {Claim:}$ Is every point of every open set $E \subset R^2$ a limit point of E? Answer the same question for closed sets in $E \subset R^2$</p>
<p>Fr... | John | 105,625 | <p>For the open set part, you are asked to show whether the following statement is correct:</p>
<p>Let $E$ be an open set in $\mathbb{R}^2$. If $p\in E$, then $p$ is a limit point of $E$.</p>
<p>Your counterexample by letting $E=\emptyset$ is not correct. Because by definition, empty set contains no element, which me... |
2,143,510 | <p>NOTE: There are some other similar questions, but I got a negative answer to this question from my proof. Please find out the errors in my reasoning. </p>
<p>$\mathbf {Claim:}$ Is every point of every open set $E \subset R^2$ a limit point of E? Answer the same question for closed sets in $E \subset R^2$</p>
<p>Fr... | DanielWainfleet | 254,665 | <p>You can do this more easily by noting that $\mathbb R^2$ has no isolated points. A point $p$ in a space $X$ is isolated iff $\{p\}$ is open. </p>
<p>Let $p\in U\subset R^2$ where $U$ is open. If $V$ is any nbhd of $p,$ then there exists open $V'$ with $p\in V'\subset V.$ </p>
<p>Now $U\cap V'$ is open and not ... |
2,084,624 | <p>The question is : </p>
<p>Is $\sum_{k=1}^\infty \frac{(-3)^k(k!)}{k^k}$ convergent? </p>
<p>Note : I can't find the limit of its main term. I know the answer must be related to some test about convergence of series ... I don't know which one and i can't find the limit.</p>
| Behrouz Maleki | 343,616 | <p>Apply <a href="https://en.wikipedia.org/wiki/Stirling's_approximation" rel="nofollow noreferrer">Stirling's approximation</a>
$$k!\sim \sqrt{2k\pi}\left(\frac{k}{e}\right)^k$$
and use <a href="https://en.wikipedia.org/wiki/Root_test" rel="nofollow noreferrer">Root Test</a>.</p>
|
148,807 | <p>I'm not sure if these types of questions are accepted here or not (I'm very sorry if it's not), but it would be great if anyone could explain me this.</p>
<blockquote>
<p><strong>Question:</strong>
Using his bike, Daniel can complete a paper route in 20 minutes. Francisco, who walks the route, can complete it i... | Inquest | 35,001 | <p>Daniel finishes $\frac{1}{20}$ th of his work in 1 min. Fransico finishes $\frac{1}{30}$ th of his work in 1 min. In one min (simultaneously), they finish off $\frac{1}{30}$ + $\frac{1}{20}$ = $\frac{1}{12}$ of the work (Assuming no dependency which is true in this case as they are starting from opposite ends). So, ... |
3,397,548 | <p>For a sequence <span class="math-container">$\{x_n\}_{n=1}^{\infty}$</span>, define <span class="math-container">$$\Delta x_n:=x_{n+1}-x_n,~\Delta^2 x_n:=\Delta x_{n+1}-\Delta x_n,~(n=1,2,\ldots)$$</span> which are named <strong>1-order</strong> and <strong>2-order difference</strong>, respectively. </p>
<p>The pro... | Martin R | 42,969 | <p>Yes: If <span class="math-container">$(x_n)$</span> is bounded and <span class="math-container">$\lim_{n \to \infty}\Delta^2 x_n = 0$</span> then <span class="math-container">$\lim_{n \to \infty}\Delta x_n = 0$</span>. That is a consequence of the following general estimate:</p>
<blockquote>
<p>If <span class="ma... |
122,945 | <p>Let $f:S^n\to C$ be a continuous function, $n\geq 1$. When $n=1$, this is a well-known theorem, called Kellog's theorem (or sometimes Kellog-Warschawski's theorem) which states the following</p>
<p>Theorem: Fix $k \geq 0, 0<\alpha<1$. Let $f\in C^{k,\alpha}(S^1)$. Then its harmonic extension $H(f)$, which is ... | Nagaraj Iyengar | 167,228 | <p>In respect of your main question, the answer is yes.
Please refer to the Algebraic Lemma on pp378 of the article
C^(1/,1/3)- regularity in the Dirichlet problem,
available at: <a href="https://www.sciencedirect.com/science/article/pii/S0898122107001927" rel="nofollow noreferrer">https://www.sciencedirect.com/science... |
3,811,753 | <p>Show that the equation:</p>
<p><span class="math-container">$$
y’ = \frac{2-xy^3}{3x^2y^2}
$$</span></p>
<p>Has an integration factor that depends on <span class="math-container">$x$</span> And solve it that way.</p>
<hr />
<p>Already we got to:</p>
<p><span class="math-container">$$
y’ + \frac{xy^3}{3x^2y^2} = \fra... | user577215664 | 475,762 | <p><span class="math-container">$$y’ = \frac{2-xy^3}{3x^2y^2}$$</span>
Is linear if you substitute <span class="math-container">$w=y^3$</span> and <span class="math-container">$w'=3y^2y'$</span>
<span class="math-container">$$3y^2y’ = \frac{2-xy^3}{x^2}$$</span>
<span class="math-container">$$w' = \frac{2-xw}{x^2}$$</s... |
230,504 | <p>Again, this question is related (**) to a <a href="https://mathoverflow.net/questions/101700/large-cardinals-without-the-ambient-set-theory?rq=1">previous one</a>:</p>
<p>in standard books on basic set theory, after stating the axioms of ZFC, ordinal numbers are introduced early on. Afterwards cardinals appear: t... | Mohammad Golshani | 11,115 | <p>I would like to suggest the theory of "<a href="https://books.google.com/books/about/Ordinal_algebras.html?id=cC44AAAAMAAJ">ordinal algebras</a>" and "<a href="https://books.google.com/books?id=9pM6AAAAMAAJ&q=inauthor:%22Alfred+Tarski%22+cardinal&dq=inauthor:%22Alfred+Tarski%22+cardinal&hl=en&sa=X&am... |
3,350,021 | <blockquote>
<p>We have the following quadratic equation:</p>
<p><span class="math-container">$2x^2-\sqrt{3}x-1=0$</span> with roots <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span>.</p>
<p>I have to find <span class="math-container">$x_1^2+x_2^2$</span> and <span clas... | Quanto | 686,284 | <p>Take the step below,</p>
<p><span class="math-container">$$(x_1-x_2)^2= x_1^2+x_2^2 - 2x_1x_2=\dfrac{7}{4}-2(-\frac 12) = \frac{11}{4}
$$</span></p>
<p>Thus, </p>
<p><span class="math-container">$$|x_1-x_2|=\frac{\sqrt{11}}{2}$$</span></p>
|
3,413,364 | <blockquote>
<p>Consider the set of points <span class="math-container">$$O = \{ x \in P \mid \alpha^* = C^T x \}$$</span> where <span class="math-container">$P \subseteq \mathbb R^n$</span> is a closed convex set, <span class="math-container">$C \in \mathbb R^n$</span> and <span class="math-container">$\alpha^* = \m... | gerw | 58,577 | <p>You have
<span class="math-container">$$ O = P \cap \{x \in \mathbb R^n | \alpha^* = C^\top x\},$$</span>
i.e., it is an intersection of two closed sets. Hence it is closed.</p>
|
227,109 | <p>I keep mixing them up, because they are very similar.</p>
<p>Some contrapositives resemble some contradictions.</p>
| ncmathsadist | 4,154 | <p>The contrapositive says that to argue $P\implies Q$, you instead argue $\sim Q\implies \sim P$.</p>
<p>Argument by contradiction is done by assuming $P$ and showing $P \implies
\rm{False}$. </p>
<p>Proving there is an infinity of primes is done by contradiction. You assume that there are finitely many. You take ... |
543,712 | <p>I am stuck on the following problem that says:</p>
<blockquote>
<p>Let <span class="math-container">$p,q$</span> be 2 complex numbers with <span class="math-container">$|p|<|q|$</span>. Let <span class="math-container">$$f(z)=\sum\{3p^n-5q^n\}z^n$$</span> Then the radius of convergence of <span class="math-conta... | angryavian | 43,949 | <p>Yes, there are $P(13,5)$ ways to choose five people to line up <em>in a certain order</em> to form the committee. But you count each committee $5!$ times (for example {Amy, Bob, Carl, Doug, Ed} is the same as {Ed, Doug, Carl, Bob, Amy}, but you counted them as being different). So we need to divide by $5!$. By the w... |
543,712 | <p>I am stuck on the following problem that says:</p>
<blockquote>
<p>Let <span class="math-container">$p,q$</span> be 2 complex numbers with <span class="math-container">$|p|<|q|$</span>. Let <span class="math-container">$$f(z)=\sum\{3p^n-5q^n\}z^n$$</span> Then the radius of convergence of <span class="math-conta... | Arcane | 160,050 | <p><a href="https://math.stackexchange.com/questions/567606/when-to-use-permutations-or-combinations/849150#849150">When to use Permutations or Combinations</a></p>
<p>I've also illustrated the concept here. The basic difference is to understabd you have to just choose a community (i.e. selection) and not prepare a Se... |
187,975 | <p>Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states that for every $t \in [0,1]$, there exists a set $S \in \Sigma$ with $\mu(S) = t$.</p>
<p>I would like to use the foll... | Salvo Tringali | 16,537 | <p>It is also a special case of Theorem 15 (p. 43) in:</p>
<blockquote>
<p>A. Fryszkowski, <em>Fixed Point Theory for Decomposable Sets</em>, Topological Fixed Point Theory and Its Applications <strong>2</strong>, Dordrecht: Kluwer Academic Publishers, 2004.</p>
</blockquote>
|
331,962 | <p>We have an first order ODE : </p>
<p>Equation1 : $y' + y = x$ ?
We can view the left-hand side as an operator acting on $y$. </p>
<p>In that case $L=(d/dx + 1)$ </p>
<p>$L(y_1) = x$<br>
$L(y_2)=x$<br>
$L(y_1+y_2)=x$<br>
So, clearly $L(y_1+y_2) = x \neq L(y_1)+L(y_2) = 2x$ </p>
<p>So why is $y'+y=x$ ... | nerdy | 66,708 | <p>Try to solve my confusion, please. </p>
<p>L[y1] = y1' + y1 </p>
<p>L[y2] = y2' + y2 </p>
<p>In this case :<br>
L[y1+y2] = (y1+y2)' + y1 + y2 = L[y1] + L[y2] </p>
<p>At the same time, by the ODE :<br>
L[y] = x </p>
<p>So, L[y1]=L[y2]=x<br>
L[y1+y2] = x as well </p>
<p>So, L[y1]+L[y2] =/... |
3,294,123 | <p>Define <span class="math-container">$f(x)=x^{-1}(\log x)^{-2}$</span> if <span class="math-container">$0<x<\frac{1}{2}$</span>, <span class="math-container">$f(x)=0$</span> on the rest of <span class="math-container">$R$</span>. Then <span class="math-container">$f \in L^1(R)$</span>. Show that
<span class="m... | Matematleta | 138,929 | <p>The other answer is the slick way to go, but the hint reduces the problem to a calculation:</p>
<p>Take <span class="math-container">$0<x<1/4$</span> and <span class="math-container">$r=x$</span>. Then, </p>
<p><span class="math-container">$Mf(x)\ge \frac{1}{B_r(x)}\int_{B_r(x)} |f|=\frac{1}{2x}\int_{0}^{2x}... |
43,956 | <p>There is this example at the Wikipedia article on Quotient spaces (QS):</p>
<blockquote>
<p>Consider the set $X = \mathbb{R}$ of all real numbers with the ordinary topology, and write $x \sim y$ if and only if $x−y$ is an integer. Then the quotient space $X/\sim$ is homeomorphic to the unit circle $S^1$ via the h... | C-star-W-star | 79,762 | <p>The quotient topology is tricky! ...</p>
<p><strong>There's a number of aspects to consider; all of them basically boil down to:</strong></p>
<ol>
<li>Equip the quotient topology: $\pi:(X,\mathcal{S})\to Y$</li>
<li>Check for quotient map: $\pi:(X,\mathcal{S})\to (Y,\mathcal{T})$</li>
</ol>
<p><strong>1.) Why do ... |
742,216 | <p>$a$, $b$, $c$, $d$ are rational numbers and all $> 0$.</p>
<p>$\max \left\{\dfrac{a}{b} , \dfrac{c}{d}\right\} \geq \dfrac{a+c}{b+d}\geq \min
\left\{\dfrac{a}{b} , \dfrac{c}{d}\right\}$</p>
<p>Hope someone can help me with this one. How would you go on proving the validity? Thanks in advance.</p>
| Martin Sleziak | 8,297 | <p>Let $m=\min\{a/b,c/d\}$. This means that
$$ m \le \frac ab \qquad \text{and} \qquad m\le \frac cd,$$
which is equivalent to $bm\le a$ and $dm\le c$. Thus we get
$$a+c\ge bm+dm=(b+d)m$$
which is equivalent to
$$\frac{a+c}{b+d}\ge m.$$</p>
<p>The proof of the inequality for maximum is similar.</p>
<p>(Note that in... |
55,404 | <p>I have been searching for a version of the isoperimetric inequality which is something like:</p>
<p>$P(\Omega) - 2\sqrt{\pi} Vol(\Omega)^{1/2} \geq \pi (r_{out}^2 - r_{in}^2)$ where $r_{out}$ and $r_{in}$ are the inner and outer radius of a given set. There are of course details which I am missing such as what kind... | Mark Meckes | 1,044 | <p>A classical result along these lines is <a href="http://en.wikipedia.org/wiki/Bonnesen%27s_inequality" rel="nofollow">Bonnesen's inequality</a>, which states
$$
L^2 - 4\pi A \ge \pi^2 (r_{out} - r_{in})^2,
$$
where $L$ is the length and $A$ is the enclosed area of a simple planar closed curve. There are many other ... |
670,813 | <p>Let $Y$ be a closed subspace of a compact space $X$. Let $i:Y \to X$ the inclusion and $r:X \to Y$ a retraction ($r \circ i = Id_Y$). I have to prove that exists this short exact sequence
$$ 0 \to K(X,Y) \to K(X) \to K(Y) \to 0.$$
Then I have to verify that $K(X) \simeq K(X,Y) \oplus K(Y)$. How can I do it? I thin... | Georges Elencwajg | 3,217 | <p>This is purely formal and relies on the fact that $K$ is a contravariant functor from topological spaces to abelian groups (actually to commutative rings but this is not needed here).<br>
Since $r\circ i=Id_Y$ we get $i^*\circ r^*=Id_{K(Y)}$ so that $r^*:K(Y)\to K(X)$ is a section of $i^*:K(X)\to K(Y)$ and your exac... |
3,288,010 | <p>The following snippet is from Adamek, Rosicky:Algebra and local presentability,how algebraic are.</p>
<p>It is unclear to me the end of Example 5.1:</p>
<p>Since <span class="math-container">$e$</span> is the coequalizer of <span class="math-container">$\bar{u}_1,\bar{u}_2$</span> in <span class="math-container">$... | lhf | 589 | <p>Another approach is simply to compute <span class="math-container">$x^2+2y^2 \bmod 8$</span> for <span class="math-container">$x,y =0, \dots 7$</span>. The result is <span class="math-container">$0, 1, 2, 3, 4, 6$</span>. Therefore, if <span class="math-container">$z \equiv 5,7 \bmod 8$</span>, then <span class="mat... |
7,575 | <p>How could I display text that flashed red for a half second or so and then reverted to black? (Or was put in bold and reverted to normal, etc.)</p>
| Szabolcs | 12 | <p>It's worth pointing out that several of the other answers keep dynamically re-evaluating an expression until infinity.</p>
<p>Here's a solution that fades out the text gradually, and more importantly: it does not keep re-evaluating the <code>Dynamic</code> expression until infinity. The third argument of <code>Clo... |
7,575 | <p>How could I display text that flashed red for a half second or so and then reverted to black? (Or was put in bold and reverted to normal, etc.)</p>
| Alexei Boulbitch | 788 | <p>As a variation of answer to this question, here is an approach I used for lectures. Here are the functions Ac and Pl that either accentuate or leave the text plain:</p>
<pre><code>Ac[expr_] :=
DynamicModule[{c1 = 0}, EventHandler[
Dynamic[
If[c1 == 0,
Style[expr, Black, Plain, 22, Italic] // Expres... |
20,567 | <p>Excuse me if the language is a bit off I'm not a native English speaker. I've been studying(self study) computers and programming for a little over two years, but do not have much education. I've been taking a math class for the last 11 weeks and start an new one next week. I've been using Unix/Linux and Stackoverfl... | mrf | 19,440 | <p>Questions at every level are welcome.</p>
<p>Just make sure to provide enough detail and context to your question, preferably pinpointing what you are having problems with. Don't just copy-paste textbook assignment.</p>
<p>Have a look, for example at the questions tagged <a href="https://math.stackexchange.com/que... |
4,008,152 | <p>Question itself: Throw a coin one million times. What is the expected number of sequences of six tails, if we <strong>do not allow overlap</strong>?</p>
<p>I know when overlap is allowed, the answer is (1,000,000-5)/(2^6). Not sure if we can just do (1,000,000-5)/(2^6) divided by 6 if overlap is not allowed?</p>
<p>... | user334639 | 221,027 | <p>I think I can compute that with an error of plus or minus 1.</p>
<p>This is a sketchy argument that you can make rigorous using Ergodic Theory or Palm measures.</p>
<p>Let us group runs of T from left to right, so a run of 14 T's has a run of 6 starting at position 1, another run starting at position 7, and two sing... |
4,008,152 | <p>Question itself: Throw a coin one million times. What is the expected number of sequences of six tails, if we <strong>do not allow overlap</strong>?</p>
<p>I know when overlap is allowed, the answer is (1,000,000-5)/(2^6). Not sure if we can just do (1,000,000-5)/(2^6) divided by 6 if overlap is not allowed?</p>
<p>... | Sal Elder | 1,025,101 | <p>We can break this into two problems. First, what's the expected number of sequences of <span class="math-container">$T^{6k}$</span> without a preceding <span class="math-container">$H$</span>? (I.e., number of <span class="math-container">$HT^{6k}$</span> in any position, or <span class="math-container">$T^{6k}$</sp... |
4,008,152 | <p>Question itself: Throw a coin one million times. What is the expected number of sequences of six tails, if we <strong>do not allow overlap</strong>?</p>
<p>I know when overlap is allowed, the answer is (1,000,000-5)/(2^6). Not sure if we can just do (1,000,000-5)/(2^6) divided by 6 if overlap is not allowed?</p>
<p>... | user1122507 | 1,122,507 | <p>I was inspired by Sal Elder, but I think there are some problems with his answer.
Below is my answer:</p>
<p>First step: calculate <span class="math-container">$E_k$</span>:the expected number of sequences of more than 6k' T with a preceding H(or at the begining).</p>
<p>Second step: sum over <span class="math-conta... |
18 | <p>Some teachers make memorizing formulas, definitions and others things obligatory, and forbid "aids" in any form during tests and exams. Other allow for writing down more complicated expressions, sometimes anything on paper (books, tables, solutions to previously solved problems) and in yet another setting students a... | Community | -1 | <p>In many cases, there is a choice between memorizing and understanding. E.g., a calc student can memorize that <span class="math-container">$\cos'=-\sin$</span>, or they can understand that it's a phase shift of 90 degrees and figure out the direction of the phase shift by visualization. Understanding is better, but ... |
743,227 | <p>I've this question:</p>
<blockquote>
<p>Find the area of the intersection between the sphere $x^2 + y^2 + z^2 = 1$ and the cylinder $x^2 + y^2 - y = 0$.</p>
</blockquote>
<p>Is this second equation even a closed shape? If one were to plot points satisfying that equation, one gets things like $(2, \sqrt{-2})$, $(... | Mark Bennet | 2,906 | <p>The equation $x^2+y^2-y=0$ can be rewritten $x^2+(y-\frac 12)^2=\cfrac 14$.</p>
<p>For any value of $z$ this is a circle, so you should be able to see how this makes the figure a cylinder (like a straight line, a cylinder in this terminology has no ends).</p>
|
98,700 | <blockquote>
<p>Suppose you wanted to write the number 100000. If you type it in ASCII, this would take 6 characters (which is 6 bytes). However, if you represent it as unsigned binary, you can write it out using 4 bytes.</p>
</blockquote>
<p>(from <a href="http://www.cs.umd.edu/class/sum2003/cmsc311/Notes/BitOp/asc... | JRN | 18,398 | <p>This is more of a computer science/engineering question than a math question.</p>
<p>Look at <a href="http://www.cs.umd.edu/class/sum2003/cmsc311/Notes/Data/unsigned.html" rel="nofollow">http://www.cs.umd.edu/class/sum2003/cmsc311/Notes/Data/unsigned.html</a>. It asks you to "assume that a typical unsigned int use... |
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